I am reading Super-Golden-Gates for $PU(2)$ by Ori Parzanchevski and Peter Sarnak. If I understand correctly then in section 4.1.3 they seem to be saying that the matrices $$ F:= \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1\\ i & -i \end{bmatrix}, S:=\zeta_8^7 \begin{bmatrix} 1 & 0\\ 0 & i \end{bmatrix}, \tau_{24}:= \frac{1}{\sqrt{-10+2\sqrt{2}}} \begin{bmatrix} -1-\sqrt{2} & 2-\sqrt{2}+i\\ 2-\sqrt{2}-i & 1+\sqrt{2} \end{bmatrix} $$ generate an S-arithemtic group. And in section 4.1.4 they seem to be saying that the matrices $$ F:= \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1\\ i & -i \end{bmatrix}, \Phi:=\frac{i}{2} \begin{bmatrix} 1 & \phi-i/\phi \\ \phi+i/ \phi & -1 \end{bmatrix}, \tau_{60}:= \frac{i}{\sqrt{7+5\phi}} \begin{bmatrix} 2+\phi & 1-i\\ 1+i & -2-\phi \end{bmatrix} $$ also generate an $ S $-arithmetic group (here $ \phi=\frac{1+\sqrt{5}}{2} $ denotes the golden ratio).
In general is there a good way to check if a given set of matrices generates an $ S $-arithmetic group?
That is, let $ g_1, \dots, g_k $ be finite order matrices in $ SU(2) $. Is there any easy way to check if the group they generate $ \Gamma:= <g_1, \dots, g_k> $ is an $ S $-arithmetic group?
It is a necessary condition that $ \Gamma $ should be infinite and moreover dense in $ SU(2) $. Are there any other necessary conditions that are easy to check?
Is there some way in GAP or Sage to check if a given set of finite order matrices generates an $ S $-arithmetic group?